Abstract

In this paper, the demiclosed principle for a k-asymptotically strictly pseudononspreading mapping is shown. Meanwhile, an iterative scheme is introduced to approximate a common element of the set of common fixed points of k-asymptotically strictly pseudononspreading mappings and the set of solutions of mixed equilibrium problems in Hilbert spaces, and some weak and strong convergence theorems are proved. The results presented in this paper improve and extend some recent corresponding results.

MSC:47H09, 47J25.

Keywords

1 Introduction

Let H be a real Hilbert space with the inner product 〈⋅,⋅〉 and the norm ∥⋅∥. Let C be a nonempty closed convex subset of H and F:C×C→R be a bifunction, where R is the set of real numbers. The equilibrium problem (for short, EP) is to find x∗∈C such that

F(x∗,y)≥0,∀y∈C.

(1.1)

The set of solutions of EP is denoted by EP(F). Given a mapping T:C→C, let F(x,y)=〈Tx,y−x〉 for all x,y∈C. Then x∗∈EP(F) if and only if x∗∈C is a solution of the variational inequality 〈Tx,y−x〉≥0 for all y∈C, i.e., x∗ is a solution of the variational inequality.

Let φ:C→R∪{+∞} be a function. The mixed equilibrium problem (for short, MEP) is to find x∗∈C such that

F(x∗,y)+φ(y)−φ(x∗)≥0,∀y∈C.

(1.2)

The set of solutions of MEP is denoted by MEP(F,φ).

If φ=0, then mixed equilibrium problem (1.2) reduces to (1.1).

If F=0, then mixed equilibrium problem (1.2) reduces to the following convex minimization problem:

Find x∗∈C such that φ(y)≥φ(x∗),∀y∈C.

(1.3)

The set of solutions of (1.3) is denoted by CMP(φ).

The mixed equilibrium problem (MEP) includes several important problems arising in physics, engineering, science optimization, economics, transportation, network and structural analysis, Nash equilibrium problems in noncooperative games and others. It has been shown that variational inequalities and mathematical programming problems can be viewed as a special realization of abstract equilibrium problems (e.g., [1, 2]). Many authors have proposed some useful methods to solve the EP, MEP; see, for instance, [1–8] and the references therein.

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. Following Kohsaka and Takahashi [9–11], a mapping T:C→C is said to be nonspreading if

2∥Tx−Ty∥2≤∥Tx−y∥2+∥Ty−x∥2for all x,y∈C.

It is easy to see that the above inequality is equivalent to

∥Tx−Ty∥2≤∥x−y∥2+2〈x−Tx,y−Ty〉for all x,y∈C.

In 1967, Browder and Petryshyn [12] introduced the concept of k-strictly pseudononspreading mapping.

Let H be a real Hilbert space. A mapping T:D(T)⊂H→H is said to be k-strictly pseudononspreading if there exists k∈[0,1) such that

∥Tx−Ty∥2≤∥x−y∥2+k∥x−Tx−(y−Ty)∥2+2〈x−Tx,y−Ty〉,∀x,y∈D(T).

Clearly, every nonspreading mapping is k-strictly pseudononspreading.

In 2012, Osilike [13] introduced a class of nonspreading type mappings, which is more general than the mappings studied in [14] in Hilbert spaces, and proved some weak and strong convergence theorems in real Hilbert spaces. Recently, Chang [15] studied the multiple-set split feasibility problem for asymptotically strict pseudocontraction in the framework of infinite-dimensional Hilbert spaces.

Let H be a real Hilbert space. A mapping T:D(T)⊂H→H is said to be a k-asymptotically strict pseudocontraction if there exist a constant k∈[0,1) and a sequence {kn}⊂[1,∞) with kn→1 (n→∞) such that

∥Tnx−Tny∥2≤kn∥x−y∥2+k∥x−Tnx−(y−Tny)∥2

holds for all x,y∈D(T).

Definition 1.3 Let C be a nonempty subset of a real Hilbert space H. A mapping T:C→C is said to be k-asymptotically strictly pseudononspreading if there exist a constant k∈[0,1) and a sequence {kn}⊂[1,∞) with kn→1 (n→∞) such that

∥Tnx−Tny∥2≤kn∥x−y∥2+k∥x−Tnx−(y−Tny)∥2+2〈x−Tnx,y−Tny〉,∀x,y∈C.

(1.4)

It is easy to see that the class of k-asymptotically strictly pseudononspreading mappings is more general than the classes of k-strictly pseudononspreading mappings and k-asymptotically strict pseudocontractions.

Example 1.4 Let X=l2 with the norm ∥⋅∥ defined by

∥x∥=∑i=1∞xi2,∀x=(x1,x2,…,xn,…)∈X,

and C={x=(x1,x2,…,xn,…)|xi∈R1,i=1,2,…} be an orthogonal subspace of X (i.e., ∀x,y∈C, we have 〈x,y〉=0). It is obvious that C is a nonempty closed convex subset of X. For each x=(x1,x2,…,xn,…)∈C, we define the mapping T:C→C by

Tx={(x1,x2,…,xn,…)if ∏i=1∞xi<0;(−x1,−x2,…,−xn,…)if ∏i=1∞xi≥0.

(1.5)

Next we prove that T is a k-asymptotically strictly pseudononspreading mapping.

In fact, for any x,y∈C.

Case 1. If ∏i=1∞xi<0 and ∏i=1∞yi<0, then we have Tnx=x, Tny=y, and so inequality (1.4) holds for any k∈[0,1).

Case 2. If ∏i=1∞xi<0 and ∏i=1∞yi≥0, then we have that Tnx=x, Tny=(−1)ny. This implies that

Thus inequality (1.4) still holds for any k∈[0,1). Therefore the mapping defined by (1.5) is a k-asymptotically strictly pseudononspreading mapping.

A mapping T:C→C is said to be uniformly L-Lipschitzian if there exists a constant L>0 such that for all (x,y)∈H×H,

∥Tnx−Tny∥≤L∥x−y∥.

(1.6)

A Banach space E is said to satisfy Opial’s condition if, for any sequence {xn} in E, xn⇀x implies that lim supn→∞∥xn−x∥<lim supn→∞∥xn−y∥ for all y∈E with y≠x. It is well known that every Hilbert space satisfies Opial’s condition.

A mapping T with domain D(T) and range R(T) in E is said to be demiclosed at p if whenever {xn} is a sequence in D(T) such that {xn} converges weakly to x∗∈D(T) and {Txn} converges strongly to p, then Tx∗=p.

T is said to be semi-compact if for any bounded sequence {xn}⊂H with limn→∞∥xn−Txn∥=0, there exists a subsequence {xni} of {xn} such that {xni} converges strongly to a point x∗∈H.

Recently, Zhao and Chang [16] proposed the following algorithm for solving k-strictly pseudononspreading mappings and equilibrium problem in Hilbert spaces.

{F(un,y)+1rn〈y−un,un−xn〉≥0,∀y∈C,xn+1=α0,nun+∑i=1∞αi,nSi,βun,

(1.7)

where Si,β:=βI+(1−β)Si, αi,n⊂(0,1). Under some suitable conditions, they proved that the sequences {xn}, {yn} weakly and strongly converge to a solution of the problem x∗∈⋂i=1∞F(Si)∩EP(F).

For finding a split feasibility problem for k-strictly pseudononspreading mappings in a Hilbert space, in [17], Quan and Chang presented the following iterative method:

where γ is a constant and γ∈(0,1−κλ), λ is the spectral of the operator A∗A, κ=max{κ1,κ2,…,κN}, and {αn} is a sequence in (0,1−ϱ] with ϱ=max{ϱ1,ϱ2,…,ϱN}. Under some suitable conditions, they proved that {xn} weakly and strongly converges to a split fixed point x∗∈Γ.

Inspired and motivated by the recent works of Zhao and Chang [16], Quan and Chang [17], etc., in this paper, we propose an iterative scheme to approximate a common element of the set of solutions of k-asymptotically strictly pseudononspreading mappings and mixed equilibrium problem in infinite-dimensional Hilbert spaces. Some weak and strong convergence theorems are proved. At the same time, the demiclosed principle of a k-asymptotically strictly pseudononspreading mapping is shown. The results presented in this paper improve and extend some recent corresponding results.

2 Preliminaries

Throughout this paper, we denote the strong convergence and weak convergence of a sequence {xn} to a point x∈X by xn→x, xn⇀x, respectively.

Let H be a Hilbert space with the inner product 〈⋅,⋅〉 and the norm ∥⋅∥, let C be a nonempty closed convex subset of H. For every point x∈H, there exists a unique nearest point of C, denoted by PCx, such that ∥x−PCx∥≤∥x−y∥ for all y∈C. Such a PC is called the metric projection from H onto C. It is well known that PC is a firmly nonexpansive mapping from H to C, i.e.,

∥PCx−PCy∥2≤〈PCx−PCy,x−y〉,∀x,y∈H.

Further, for any x∈H and z∈C, z=PCx if and only if

〈x−z,z−y〉≥0,∀y∈C.

(2.1)

For solving mixed equilibrium problems, we assume that the bifunction F:C×C→R satisfies the following conditions:

(A1) F(x,x)=0, ∀x∈C;

(A2) F(x,y)+F(y,x)≤0, ∀x,y∈C;

(A3) For all x,y,z∈C, limt↓0F(tz+(1−t)x,y)≤F(x,y);

(A4) For each x∈C, the function y↦F(x,y) is convex and lower semi-continuous.

The demiclosed principle and the closeness and convexity of the set of fixed points of a nonlinear mapping play very important roles in investigating many nonlinear problems. We now show the demiclosed principle of k-asymptotically strictly pseudononspreading mapping and the closeness and convexity of the set of fixed points of such a mapping, respectively.

Lemma 2.3LetCbe a nonempty closed convex subset of a real Hilbert spaceHand letT:C→Cbe a continuousk-asymptotically strictly pseudononspreading mapping. IfF(T)≠∅, then it is closed and convex.

Proof Let {xn}n=1∞⊂F(T) be a sequence which converges to x∈C, we show that x∈F(T).

Since kn→1 as n→∞, we obtain that limn→∞∥z−Tnz∥2=0, which implies that limn→∞Tnz=z, z=limn→∞Tnz=Tlimn→∞(Tn−1z)=Tz. Hence, z∈F(T), which means that F(T) is convex. □

Lemma 2.4LetCbe a nonempty closed convex subset of a real Hilbert spaceH, and letT:C→Cbe ak-asymptotically strictly pseudononspreading and uniformlyL-Lipschitzian mapping. Then, for any sequence{xn}inCconverging weakly to a pointpand{∥xn−Txn∥}converging strongly to 0, we havep=Tp.

Proof Since limn→∞∥xn−Txn∥=0, by induction we can prove that

limn→∞∥xn−Tmxn∥=0for each m≥1.

In fact, it is obvious that the conclusion is true for m=1. Suppose that the conclusion holds for m>1, now we prove that the conclusion is also true for m+1.

where{αn}is a sequence in(0,1)withlim infn→∞αn>0, {βn}is a sequence in(0,1−k)withlim infn→∞βn>0, k=max{τ1,τ2,…,τN}∈(0,1), and the sequence{rn}⊂(0,∞)satisfies thatlim infn→∞rn>0andlimn→∞|rn+1−rn|=0. IfΓ:=⋂i=1NF(Si)⋂i=1NF(Ti)∩MEP(F,φ)≠∅, then the sequence{xn}converges weakly to a pointx∗∈Γ.

Proof The proof is divided into four steps.

Step 1. Firstly, we prove that limn→∞∥xn−p∥ exists for any p∈Γ.

Taking p∈Γ and putting ρ=max{l1,l2,…,lN}∈(0,1), it follows from Lemma 2.1 that un=Trnxn, p=Trnp, we have

Since limn→∞∥xn−p∥ exists for any p∈Γ and ∥xn−p∥−∥xn−yn∥≤∥yn−p∥≤∥xn−p∥+∥xn−yn∥, it follows from (3.36) that limn→∞∥yn−p∥=limn→∞∥xn−p∥ holds. Similarly, limn→∞∥un−p∥=limn→∞∥xn−p∥ holds for any p∈Γ.

Step 3. We show that x∗∈Γ:=⋂i=1NF(Si)⋂i=1NF(Ti)∩MEP(F,φ).

Firstly, we show that x∗∈⋂i=1NF(Si)⋂i=1NF(Ti).

In fact, since {yn} is bounded, there exists a subsequence {yni}⊂{yn} such that {yni}⇀x∗∈C. Hence, for any positive integer j=1,2,…,N, there exists a subsequence {ni(j)}⊂{ni} with ni(j)(modN)=j such that {yni(j)}⇀x∗. Again, by (3.34) we know that ∥yiN+j−SjuiN+j∥→0 as i→∞, therefore we have that limni(j)→∞∥yni(j)−Sjyni(j)∥=0.

Since Sj is demiclosed at zero, it follows from Lemma 2.4 that x∗∈F(Sj). By the arbitrariness of j=1,2,…,N, we have

x∗∈⋂i=1NF(Si).

On the other hand, since limn→∞∥yn−un∥=0, we know that uni⇀x∗, too. Similarly, it follows from (3.33) and Lemma 2.4 that x∗∈F(Tj). By the arbitrariness of j=1,2,…,N, we have

x∗∈⋂i=1NF(Ti).

Now, we show that x∗∈MEP(F,φ).

By Lemma 2.1, since un=Trnxn, we have

F(un,y)+φ(y)−φ(un)+1rn〈y−un,un−xn〉≥0,∀y∈K.

(3.38)

From (A2), we obtain

φ(y)−φ(un)+1rn〈y−un,un−xn〉≥−F(un,y)≥F(y,un),

(3.39)

and hence

φ(y)−φ(uni)+1rni〈y−uni,uni−xni〉≥F(y,uni).

(3.40)

By lim infn→∞rn>0, we have limi→∞∥uni−xni∥rni=0. Since uni⇀x∗, it follows from (A4) and the weak lower semicontinuity of φ that

F(y,x∗)−φ(y)+φ(x∗)≤0.

(3.41)

Put zt=ty+(1−t)x∗ for all t∈(0,1] and y∈C. Consequently, we get zt∈C. Hence

Due to uni⇀x∗, we know that xni⇀x∗ from (3.37). Suppose that there exists another subsequence {xnj} of {xn} such that {xnj}⇀y∗∈Γ with y∗≠x∗. Using the same proof method as in Step 3, we know that y∗∈Γ. Consequently, limn→∞∥xn−y∗∥ exists. By using Opial’s property of a Hilbert space, we have

wherek=max{τ1,τ2,…,τN}∈(0,1), ρ=max{l1,l2,…,lN}∈(0,1), {αn}is a sequence in(0,1)withlim infn→∞αn>0, {βn}is a sequence in(0,1−k)withlim infn→∞βn>0and the sequence{rn}⊂(0,∞)withlim infn→∞rn>0andlimn→∞|rn+1−rn|=0. IfΓ:=⋂i=1∞F(Si)⋂i=1∞F(Ti)∩MEP(F,φ)≠∅, and there exists a positive integerjsuch thatSjis semi-compact, then the sequence{xn}converges strongly to a pointx∗∈Γ.

Proof Without loss of generality, we can assume that S1 is semi-compact. It follows from (3.34) that

∥yni(1)−S1yni(1)∥→0,ni(1)→∞.

Therefore, there exists a subsequence of {yni(1)} (for the sake of convenience we still denote it by {yni(1)}) such that yni(1)→y∗∈H1. Since yni(1)⇀y∗, x∗=y∗, and so yni(1)→x∗∈Γ. By virtue of the fact that limn→∞∥yn−p∥ exists, we know that

limn→∞∥yn−x∗∥=limn→∞∥un−x∗∥=limn→∞∥xn−x∗∥=0.

That is, {xn}, {un} and {yn} converge strongly to the point x∗∈Γ. This completes the proof. □

4 Applications

4.1 Application to a convex minimization problem

It is well known that mixed equilibrium problem (1.2) reduces to the convex minimization problem as F=0. Therefore, Theorem 3.1 can be used to solve convex minimization problem (1.3), and the following result can be directly deduced from Theorem 3.1.

wherek=max{τ1,τ2,…,τN}∈(0,1), ρ=max{l1,l2,…,lN}∈(0,1), {αn}is a sequence in(0,1)withlim infn→∞αn>0, {βn}is a sequence in(0,1−k)withlim infn→∞βn>0, and the sequence{rn}⊂(0,∞)withlim infn→∞rn>0andlimn→∞|rn+1−rn|=0. If⋂i=1NF(Si)⋂i=1NF(Ti)∩CMP(φ)≠∅, then the sequence{xn}converges weakly to a pointx∗∈⋂i=1NF(Si)⋂i=1NF(Ti)∩CMP(φ).

4.2 Application to a convex feasibility problem

The so-called convex feasibility problem for a family of mappings {Ti}i=1ω (where ω may be a finite positive integer or +∞) is to find a point of the nonempty intersection ⋂i=1ωCi, where Ci is the fixed point set of mapping Ti, i=1,2,…,ω.

In Theorem 3.1 if F=0, φ=0, then the condition ‘un∈C such that ∀y∈C, 〈y−un,un−xn〉≥0’ is equivalent to un=PC(xn). Therefore, the following result can be directly obtained from Theorem 3.1.

where{αn}is a sequence in(0,1)withlim infn→∞αn>0and{βn}is a sequence in(0,1−k)withlim infn→∞βn>0, k=max{τ1,τ2,…,τN}∈(0,1). IfΓ:=⋂i=1NF(Si)⋂i=1NF(Ti)≠∅, then the sequence{xn}converges weakly to a pointx∗∈Γ, which is a solution of the convex feasibility problem for mappings{Ti}i=1Nand{Si}i=1N.

A variational inequality problem (VIP) is formulated as a problem of finding a point x∗ with property x∗∈C, 〈Ax∗,z−x∗〉≥0, ∀z∈C. We will denote the solution set of VIP by VI(A,C). We know that given a mapping T:C→C, let F(x,y)=〈Tx,y−x〉 for all x,y∈C. Then x∗∈EP(F) if and only if x∗∈C is a solution of the variational inequality 〈Tx,y−x〉≥0 for all y∈C, i.e., x∗ is a solution of the variational inequality.

In [20], the mixed variational inequality of Browder type (VI) is shown to be equivalent to finding a point u∈C such that

〈Au,y−u〉+φ(y)−φ(u)≥0,∀y∈C.

We will denote the solution set of a mixed variational inequality of Browder type by VI(A,C,φ).

A mapping A:C→H is said to be an α-inverse-strongly monotone mapping if there exists a constant α>0 such that 〈Ax−Ay,x−y〉≥α∥Ax−Ay∥2 for any x,y∈C. Setting F(x,y)=〈Ax,y−x〉, it is easy to show that F satisfies conditions (A1)-(A4) as A is an α-inverse-strongly monotone mapping. Then it follows from Theorem 3.1 that the following result holds.

where{αn}is a sequence in(0,1)withlim infn→∞αn>0, {βn}is a sequence in(0,1−k)withlim infn→∞βn>0, k=max{τ1,τ2,…,τN}∈(0,1), and the sequence{rn}⊂(0,∞)satisfieslim infn→∞rn>0andlimn→∞|rn+1−rn|=0. IfΓ:=⋂i=1NF(Si)⋂i=1NF(Ti)∩VI(A,C,φ)≠∅, then the sequence{xn}converges weakly to a pointx∗∈Γ.

Declarations

Acknowledgements

The authors would like to express their thanks to the reviewers and editors for their helpful suggestions and advice. This work was supported by the National Natural Science Foundation of China (Grant No. 11361070).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to this work. Both authors read and approved the final manuscript.

Authors’ Affiliations

(1)

School of Information Engineering, The College of Arts and Sciences Yunnan Normal

(2)

College of Statistics and Mathematics, Yunnan University of Finance and Economics

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